Often in communications systems data is transmitted through a noisy medium. As a result, typically the signal that reaches the receiver is not the pure clean transmitted data, but instead a mixture of the clean transmitted data and noise that was acquired as the data passed through the transmission medium. Such noise is usually minimized in a wired communication system (e.g., Ethernet), but it more prominent in a wireless or radio communications system (e.g., WiMAX, Wi-Fi, etc.)
Often to minimize the effect of the added noise, a clean version of the data (or “codeword” in the parlance of the art) is encoded with an error correction scheme. Typically a codeword is a sequence of symbols assembled in accordance with the specific rules of a code and assigned a unique meaning. In various systems an error correction scheme may include a parity check matrix. Typically, a parity check matrix is an algebraic matrix that facilitates the recovery of the codeword or clean data from the received noisy data.